Polynomials and asymptotic constants in a resurgent problem from 't Hooft
David Broadhurst, Gergő Nemes
TL;DR
This work addresses the analytic continuation of the fractional polylogarithm $G(z)=\sum_{n=1}^{\infty}\sqrt{n}\,z^n$ for $|z|\ge1$ with a branch cut on $[1,\infty)$ by employing a bilateral representation $G(z)=\frac{\sqrt{\pi}}{2}\sum_{n=-\infty}^{\infty}(2\pi i n-\log z)^{-3/2}$ and studying its negative-axis asymptotics. It develops an exponentially improved expansion $G(-e^{u})= -\frac{2}{\pi\sqrt{u}}\sum_{n=0}^{\lfloor u/2\rfloor}\eta(2n)\Gamma(2n+\tfrac{1}{2})u^{-2n} + e^{-u}\sum_{k=0}^{\infty} P_k(x)/u^{k}$ with $x=\frac{u}{2}-\lfloor u/2\rfloor$, where the $P_k(x)$ are polynomials of degree $2k+1$ whose late-term behaviour is sinusoidal: $P_k(x)\sim \frac{1}{\sqrt{2\pi}} \sin((2k+1)C-2\pi x)\,R^{2k+1}\Gamma(k+\tfrac{1}{2})$ for large $k$. The constants $C$ and $R$ are determined explicitly by $R e^{i C}=\sqrt{-1/(2+\pi i)}$, giving $C=\frac{\pi}{2}-\frac{1}{2}\arctan\left(\frac{\pi}{2}\right)$ and $R=(4+\pi^2)^{-1/4}$. The proofs rely on terminant-function expansions, contour deformations, and residue calculations, and the results connect to Hurwitz zeta values and Fermi–Dirac integrals, with implications for resurgence in quantum problems.
Abstract
In a recent study of the quantum theory of harmonic oscillators, Gerard 't Hooft proposed the following problem: given $G(z)=\sum_{n=1}^\infty\sqrt{n}\,z^n$ for $|z|<1$, find its analytic continuation for $|z|\ge1$, excluding a branch-cut $z\in[1,\,\infty)$. A solution is provided by the bilateral convergent sum $G(z)=\frac12\sqrtπ\sum_{n=-\infty}^\infty(2π{\rm i}n-\log(z))^{-3/2}$. On the negative real axis, $G(-{\rm e}^u)$ has a sign-constant asymptotic expansion in $1/u^2$, for large positive $u$. Optimal truncation leaves exponentially suppressed terms in an asymptotic expansion ${\rm e}^{-u}\sum_{k=0}^\infty P_k(x)/u^k$, with $P_0(x)=x-\frac23$ and $P_k(x)$ of degree $2k+1$ evaluated at $x=u/2-\lfloor u/2\rfloor$. At large $k$, these polynomials become excellent approximations to sinusoids. The amplitude of $P_k(x)$ increases factorially with $k$ and its phase increases linearly, with $P_k(x)\sim\sin((2k+1)C-2πx)R^{2k+1}Γ(k+\frac12)/\sqrt{2π}$, where $C\approx1.0688539158679530121571$ and $R\approx0.5181839789815558726739$ are asymptotic constants satisfying $R\exp({\rm i}\,C)=\sqrt{-1/(2+π{\rm i})}$.